Buffon`s Needle Problem

Buffon’s Needle Problem – By Arlinda Rezhdo November 30, 2011 INTRODUCTION:
Buffon, a French naturalist in the XVIII centaury introduced a probability question for
which one would have to be looking at different scenarios. “[The] problem asks to find
the probability that a needle of length l will land on a line, given a floor with equally
spaced parallel lines at distance d apart.” 1This problem has multiple conditions. The
variables we can manipulate are the length of the needle and the distance between the
lines, which are very related to each other. We then can measure the angle in which the
needle has fallen as well as if it has hit the line or not, which is what we are primarily
interested in.
SOLUTIONS TO CASES:
To solve this problem we
design different scenarios.
The first one and the most
straight forward is the one
when the length of the
needle is equal to the
distance between the lines:
Let L be the length of the
needle and therefore the
distance between the lines
in the floor. Suppose we
throw the needle at an angle
𝜃 considered to be between
0 and 𝜋 by symmetry (if the
angle is greater than 𝜋 ,
such as an angle 𝜋 + 𝜃, the
needle will be in the same Figure 2: Representation of Buffon’s needle problem. Length of the position as when thrown needle and distance can vary and so can the angle between them. with
an
angle
𝜃
itself). We
can draw a
line
parallel to
the lines
on
the Figure 1: Graph of function 𝑳𝒔𝒊𝒏𝜽/𝟐 where L = 1, bounded by 𝒚 = 𝟏/𝟐 and y = 0, and 𝒙 = 𝝅 and x=0 lines, which represent respectively the half the distance b etween lines floor,
and angle of needle. passing
through the center of the needle. Let d be the distance from the center of the needle to the
1 Wolfram MathWorld, Buffon’s Needle Problem, http://mathworld.wolfram.com/BuffonsNeedleProblem.html, access. Nov 20, 2011 Buffon’s Needle Problem – By Arlinda Rezhdo November 30, 2011 closest line. As shown in Figure 1, if 𝑑 ≤ 𝑥 the needle hits the line, but if this is not true
!
the needle does not hit the line. By trigonometry we can see that 𝑥 = ! 𝑠𝑖𝑛𝜃. We need to
!
find the probability of that occurring. If we plot the graph of 𝑑 = ! 𝑠𝑖𝑛𝜃, for the specified
values of 𝜃, we can see that for the values under the sin curve d is smaller than the
function and therefore the needle hits the curve. From this we can find the probability of
the needle hitting the line by taking the ratio of the area under the curve with respect to
!
the rectangle ! ×π, which represents the total amount of possible cases. We find this to
be:
!𝐿
𝐿
𝑠𝑖𝑛𝜃𝑑𝜃 2 (− cos 𝜃) !! −𝑐𝑜𝑠𝜋 − (−𝑐𝑜𝑠0) 1 + 1
𝐿
! 2
𝑃 𝑑 ≤ 𝑠𝑖𝑛𝜃 =
=
=
=
= 2/𝜋
𝐿
𝐿
2
𝜋
𝜋
2∗𝜋
2∗𝜋
From this relation we find the probability that the needle touches the line to be 0.63662 or
63.662%.
The second scenario is when the length of the needle is smaller than the distance between
the lines:
Let L be the
length of the
needle and d
be
the
distance
between the
lines on the
floor such that Figure 3: Graph of 𝑳 𝒔𝒊𝒏𝜽 when d>L with same boundaries as the previous occasion. 𝟐
𝐿 < 𝑑 . Same
way as above we find out that the probability that the needle falls on a line is:
! 𝐿
!
( 𝑠𝑖𝑛𝜃)𝑑𝜃
𝐿 𝑠𝑖𝑛𝜃 𝑑𝜃 2𝐿 !
2𝐿
! 2
𝑃 𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 𝑤ℎ𝑒𝑛 𝐿 < 𝑑 =
=
=
𝑠𝑖𝑛𝜃𝑑𝜃 = 𝑑𝜋
𝑑 2𝜋 𝜋𝑑 !
𝜋𝑑
!
!
Notice that, when 𝐿 = 𝑑 , 𝐿/𝑑 = 1 and therefore the 𝑃(𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 𝑤ℎ𝑒𝑛 𝐿 = 𝑑) = ! ,
which is what we got in the first case.
The third and last
scenario is when the
length of the needle is
bigger
than
the
distance between the
lines:
Let L be the length of
the needle and d be
the distance between
the lines on the floor
such that 𝑑 < 𝐿 . As
Figure 4: Graph representing L<d Buffon’s Needle Problem – By Arlinda Rezhdo November 30, 2011 seen from Figure 4, we cannot calculate this expression by using the same method as in
the previous scenarios because in some cases, the ratio turns out to be greater than one,
which is impossible for a probability. Therefore, there is another way to solve this issue.
Assume L>1: We know that in order for the needle to cross the line the condition of
!
!
!
𝑑 ≤ ! 𝑠𝑖𝑛𝜃 must be fulfilled as long as ! 𝑠𝑖𝑛𝜃 ≤ ! . In other words for
0 ≤ 𝜃 ≤ arcsin
!
!
!
𝑜𝑟 arcsin (!) ≤ 𝜃 ≤ 𝜋 , and for the remaining values of 𝜃 the
!
condition is simply 𝑑 ≤ !. So we write 𝑎𝑟𝑐𝑠𝑖𝑛(1/𝐿) = 𝛼 and:
2 !𝐿
1 𝜋
𝑃 𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 𝑤ℎ𝑒𝑛 𝑑 < 𝐿 = 2 ∗
𝑠𝑖𝑛𝜃 𝑑𝜃 +
−𝛼
𝜋 ! 2
2 2
2𝐿
2𝛼
=
1 − 𝑐𝑜𝑠𝛼 + 1 −
𝜋
𝜋
Knowing that 𝑐𝑜𝑠𝛼 = cos arcsin
𝑃 𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 =
!
!
=
!
1 − !! , we can finalize:
2𝐿
1
2
1
1 − 1 − ! + 1 − arcsin
𝜋
𝐿
𝜋
𝐿
After some additional manipulations to the formulas we reached the conclusion that he
probability that the needle dropped crosses a line is in general:
2𝐿
𝑓𝑜𝑟𝐿 ≤ 𝑑
𝜋𝑑
𝑃 𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 𝑙𝑖𝑛𝑒 =
2 𝐿
𝐿 !
𝐿
−
− 1 + arcsec
𝑓𝑜𝑟 𝐿 > 𝑑
𝜋 𝑑
𝑑
𝑑
APPLICATIONS:
Buffon’s needle is one of the most interesting problems in Integral Probability. It has
been visualized in many scenarios such as solving the situation in concentric circles or in
grids of squares.
Other than having an interesting solution, this problem is considered as one of the
techniques to current important issues such as problems involving cancer cells, cell
growth and differentiation, military strategies, chromosome positioning etc. One of the
most relevant uses is in DNA sequencing. DNA sequencing theory attempts to show
analytical foundation for the natural process of DNA sequencing and gene crossing
patterns (inheritance). In this field they have to calculate the probability that they cover
every genome sequence and that’s where Buffon’s needle typical solution comes handy.
Buffon’s Needle Problem – By Arlinda Rezhdo November 30, 2011 BIBLIOGRAPHY:
Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/BuffonsNeedleProblem.html. Accessed Nov 20, 2011
Khamis, Harry. “Buffon’s Needle Problem”. Write State University. Last updated July 18, 2001
http://www.math.wright.edu/People/Harry_Khamis/buffons_needle_problem/index.htm. Accessed
Nov 20, 2011
Reese, George. “Buffon’s Needle Problem: An analysis and simulation”.
http://mste.illinois.edu/reese/buffon/#schroeder. Accessed Nov 22, 2011
Shapiro, Joel. “The Buffon Needle Problem”. Portland State University. Last updated March 16, 2009.
http://mste.illinois.edu/reese/buffon/#schroeder. Accessed Nov 19, 2011
Buffon’s Needle Problem – By Arlinda Rezhdo November 30, 2011 APPENDIX:
Figure 5: Matlab simulation of 1000 needle drops in a floor of five rectangles with lines 100 units apart. Needle length is 10 units. Part of the active component of the project. Refer to code below
function prob = buffon(t,L,n,m)
% t - distance between lines in the grid
% l - length of needle
% n - number of spaces
% m - number of trials to measure probability
% a set of n spaces defined by n+1 vertical lines t units apart.
Buffon’s Needle Problem – By Arlinda Rezhdo November 30, 2011 % 1's represent the inside, 0's represent the lines.
A = ones(t*n+1,t*n+1);
for i=0:n
A(:,1+t*i)=zeros(t*n+1,1);
end
imshow(A);
hold on;
V=zeros(2,2,m); % create a 3D array that will keep track of the
coordinates
% of the needle and the number of needles (m)
for i=1:m
% defined the needle where vector x represents one side of the needle
and
% vector y represents the other
x = randi(1+t*n,[1,2]); % generates two random points (integers) with
max
% being the size of the grid
% The second side cannot be randomly selected because we are specifying
a
% needle length. It has to be randomly selected in such a way that it
% fulfills the length requirement.
y = x;
while true,
y = randi(1+t*n,[1,2]);
if sqrt((y(1)-x(1))^2+(y(2)-x(2))^2)==L
break;
end;
end;
line([x(1),y(1)],[x(2),y(2)]);
hold on;
V(:,:,i)=[x;y];
end;
k = 0; % counts how many needles crossed at least a line in the grid
for i=1:m
for j=V(1,1,i):V(2,1,i)
if mod(j-1,t)==0 % touches the line
k = k+1;
break;
end;
end
end;
prob = k/m;
Example and result: >> prob = buffon(10,10,30,200) prob = 0.3800